Numerical Computation of Internal and External Flows, Volume 1: Fundamentals of Computational Fluid Dynamics

696 Pages·2007·13.99 MB·English

by  Charles Hirsch

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Numerical Computation of Internal
and External Flows
Volume 1
Fundamentals of Computational Fluid
Dynamics
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Numerical Computation of
Internal and External Flows
Volume 1
Fundamentals of Computational
Fluid Dynamics
Second edition
Charles Hirsch
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD
PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
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To the Memory of
Leon Hirsch
and
Czypa Zugman
my parents,
struck by destiny
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Contents
Prefacetothesecondedition xv
Nomenclature xviii
Introduction:AnInitialGuidetoCFDandtothisVolume 1
I.1 ThepositionofCFDintheworldofvirtualprototyping 1
I.1.1 TheDefinitionPhase 2
I.1.2 TheSimulationandAnalysisPhase 3
I.1.3 TheManufacturingCyclePhase 5
I.2 ThecomponentsofaCFDsimulationsystem 11
I.2.1Step1:DefiningtheMathematicalModel 11
I.2.2Step2:DefiningtheDiscretizationProcess 13
I.2.3Step3:PerformingtheAnalysisPhase 15
I.2.4Step4:DefiningtheResolutionPhase 16
I.3 Thestructureofthisvolume 18
References 20
PartI TheMathematicalModelsforFluidFlowSimulationsat
VariousLevelsofApproximation 21
1 TheBasicEquationsofFluidDynamics 27
Objectivesandguidelines 27
1.1 Generalformofaconservationlaw 29
1.1.1 ScalarConservationLaw 30
1.1.2 Convection–DiffusionFormofaConservationLaw 33
1.1.3 VectorConservationLaw 38
TheEquationsofFluidMechanics 39
1.2 Themassconservationequation 40
1.3 Themomentumconservationlaworequationofmotion 43
1.4 Theenergyconservationequation 47
1.4.1 ConservativeFormulationoftheEnergyEquation 49
1.4.2 TheEquationsforInternalEnergyandEntropy 49
1.4.3 PerfectGasModel 50
1.4.4 IncompressibleFluidModel 53
A1.5 Rotatingframeofreference 54
A1.5.1 EquationofMotionintheRelativeSystem 54
A1.5.2 EnergyEquationintheRelativeSystem 55
A1.5.3 Crocco’sFormoftheEquationsofMotion 56
A1.6 Advancedapplicationsofcontrolvolumeformulations 57
A1.6.1 LiftandDragEstimationsfromCFDResults 57
A1.6.2 ConservationLawforaMovingControlVolume 58
Summaryofthebasicflowequations 60
vi
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Contents vii
Conclusionsandmaintopicstoremember 63
References 63
Problems 63
2 TheDynamicalLevelsofApproximation 65
Objectivesandguidelines 65
2.1 TheNavier–Stokesequations 70
2.1.1 Non-uniquenessinViscousFlows 73
2.1.1.1 Marangonithermo-capillaryflowinaliquidbridge 73
2.1.1.2 Flowaroundacircularcylinder 77
2.1.2 DirectNumericalSimulationofTurbulentFlows(DNS) 83
2.2 Approximationsofturbulentflows 86
2.2.1 LargeEddySimulation(LES)ofTurbulentFlows 87
2.2.2 ReynoldsAveragedNavier–StokesEquations(RANS) 89
2.3 Thinshearlayerapproximation(TSL) 94
2.4 ParabolizedNavier–Stokesequations 94
2.5 Boundarylayerapproximation 95
2.6 Thedistributedlossmodel 96
2.7 Inviscidflowmodel:Eulerequations 97
2.8 Potentialflowmodel 98
2.9 Summary 101
References 101
Problems 103
3 TheMathematicalNatureoftheFlowEquationsandTheir
BoundaryConditions 105
Objectivesandguidelines 105
3.1 Simplifiedmodelsofaconvection–diffusionequation 108
3.1.1 1DConvection–DiffusionEquation 108
3.1.2 PureConvection 109
3.1.3 PureDiffusioninTime 110
3.1.4 PureDiffusioninSpace 111
3.2 DefinitionofthemathematicalpropertiesofasystemofPDEs 111
3.2.1 SystemofFirstOrderPDEs 112
3.2.2 PartialDifferentialEquationofSecondOrder 116
3.3 Hyperbolicandparabolicequations:characteristicsurfacesand
domainofdependence 117
3.3.1 CharacteristicSurfaces 118
3.3.2 DomainofDependence:ZoneofInfluence 120
3.3.2.1 Parabolicproblems 120
3.3.2.2 Ellipticproblems 122
3.4 Time-dependentandconservationformofthePDEs 122
3.4.1 PlaneWaveSolutionswithTimeVariable 123
3.4.2 CharacteristicsinaOne-DimensionalSpace 128
3.4.3 NonlinearDefinitions 129
3.5 Initialandboundaryconditions 130
A.3.6 Alternativedefinition:compatibilityrelations 132
A3.6.1 CompatibilityRelations 133
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viii Contents
Conclusionsandmaintopicstoremember 136
References 137
Problems 137
PartII BasicDiscretizationTechniques 141
4 TheFiniteDifferenceMethodforStructuredGrids 145
Objectivesandguidelines 145
4.1 Thebasicsoffinitedifferencemethods 147
4.1.1 DifferenceFormulasforFirstandSecondDerivatives 149
4.1.1.1 Differenceformulaforfirstderivatives 150
4.1.1.2 FDformulasforsecondderivatives 153
4.1.2 DifferenceSchemesforOne-DimensionalModelEquations 154
4.1.2.1 Linearone-dimensionalconvectionequation 154
4.1.2.2 Lineardiffusionequation 159
4.2 Multidimensionalfinitedifferenceformulas 160
4.2.1 DifferenceSchemesfortheLaplaceOperator 162
4.2.2 MixedDerivatives 166
4.3 Finitedifferenceformulasonnon-uniformgrids 169
4.3.1 DifferenceFormulasforFirstDerivatives 172
4.3.1.1 ConservativeFDformulas 173
4.3.2 AGeneralFormulation 174
4.3.3 Cell-CenteredGrids 177
4.3.4 GuidelinesforNon-uniformGrids 179
A4.4 Generalmethodforfinitedifferenceformulas 180
A4.4.1 GenerationofDifferenceFormulasforFirst
Derivatives 181
A4.4.1.1 Forwarddifferences 182
A4.4.1.2 Backwarddifferences 183
A4.4.1.3 Centraldifferences 183
A4.4.2 HigherOrderDerivatives 184
A4.4.2.1 Secondorderderivative 186
A4.4.2.2 Thirdorderderivatives 187
A4.4.2.3 Fourthorderderivatives 189
A4.5 Implicitfinitedifferenceformulas 189
A4.5.1 GeneralApproach 189
A4.5.2 GeneralDerivationofImplicitFiniteDifference
Formula’sforFirstandSecondDerivatives 191
Conclusionsandmaintopicstoremember 195
References 196
Problems 197
5 FiniteVolumeMethodandConservativeDiscretizationwithan
IntroductiontoFiniteElementMethod 203
Objectivesandguidelines 203
5.1 Theconservativediscretization 204
5.1.1 FormalExpressionofaConservativeDiscretization 208
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Contents ix
5.2 Thebasisofthefinitevolumemethod 209
5.2.1 ConditionsonFiniteVolumeSelections 210
5.2.2 DefinitionoftheFiniteVolumeDiscretization 212
5.2.3 GeneralFormulationofaNumericalScheme 213
5.3 Practicalimplementationoffinitevolumemethod 216
5.3.1 Two-DimensionalFiniteVolumeMethod 216
5.3.2 FiniteVolumeEstimationofGradients 221
A.5.4 Thefiniteelementmethod 225
A5.4.1 FiniteElementDefinitionofInterpolationFunctions 226
A5.4.1.1 One-dimensionallinearelements 228
A5.4.1.2 Two-dimensionallinearelements 231
A5.4.2 FiniteElementDefinitionoftheEquation
Discretization:IntegralFormulation 232
A5.4.3 TheMethodofWeightedResidualsorWeakFormulation 232
A5.4.4 TheGalerkinMethod 234
A5.4.5 FiniteElementGalerkinMethodfora
ConservationLaw 237
A5.4.6 SubdomainCollocation:FiniteVolumeMethod 238
Conclusionsandmaintopicstoremember 241
References 242
Problems 243
6 StructuredandUnstructuredGridProperties 249
Objectivesandguidelines 249
6.1 StructuredGrids 250
6.1.1 CartesianGrids 252
6.1.2 Non-uniformCartesianGrids 252
6.1.3 Body-FittedStructuredGrids 254
6.1.4 Multi-blockGrids 256
6.2 Unstructuredgrids 261
6.2.1 Triangle/TetrahedraCells 262
6.2.2 HybridGrids 264
6.2.3 Quadrilateral/HexahedraCells 264
6.2.4 ArbitraryShapedElements 265
6.3 Surfaceandvolumeestimations 267
6.3.1 EvaluationofCellFaceAreas 269
6.3.2 EvaluationofControlCellVolumes 270
6.4 Gridqualityandbestpracticeguidelines 274
6.4.1 ErrorAnalysisof2DFiniteVolumeSchemes 274
6.4.2 RecommendationsandBestPracticeAdviceonGrid
Quality 276
Conclusionsandmaintopicstoremember 276
References 277
PartIII TheAnalysisofNumericalSchemes 279
7 Consistency,StabilityandErrorAnalysisofNumericalSchemes 283
Objectivesandguidelines 283